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Câu hỏi: Chứng minh rằng
\(\begin{array}{l}{\cos ^2}\left( {\gamma - \alpha } \right) + {\sin ^2}\left({\gamma - \beta } \right) - 2\cos \left({\gamma - \alpha } \right)\sin \left({\gamma - \beta } \right)\\ = {\cos ^2}\left({\alpha - \beta } \right)\end{array}\)
\(\begin{array}{l}{\cos ^2}\left( {\gamma - \alpha } \right) + {\sin ^2}\left({\gamma - \beta } \right) - 2\cos \left({\gamma - \alpha } \right)\sin \left({\gamma - \beta } \right)\\ = {\cos ^2}\left({\alpha - \beta } \right)\end{array}\)
Lời giải chi tiết
Ta có
\(\begin{array}{l}{\cos ^2}\left( {\gamma - \alpha } \right) + {\sin ^2}\left({\gamma - \beta } \right)\\ = \dfrac{{1 + \cos 2\left({\gamma - \alpha } \right)}}{2} + \dfrac{{1 - \cos 2\left({\gamma - \beta } \right)}}{2}\\ = 1 + \dfrac{1}{2}\left[ {\cos 2\left({\gamma - \alpha } \right) - \cos 2\left({\gamma - \beta } \right)} \right]\\ = 1 + \sin \left({2\gamma - \alpha - \beta } \right)\sin \left({\alpha - \beta } \right)\end{array}\)
Từ đó
\(\begin{array}{l}{\cos ^2}\left( {\gamma - \alpha } \right) + {\sin ^2}\left({\gamma - \beta } \right) - 2\cos \left({\gamma - \alpha } \right)\sin \left({\gamma - \beta } \right)\sin \left({\alpha - \beta } \right)\\ = 1 + \sin \left({2\gamma - \alpha - \beta } \right)\sin \left({\alpha - \beta } \right) - 2\cos \left({\gamma - \alpha } \right)\sin \left({\gamma - \beta } \right)\sin \left({\alpha - \beta } \right)\\ = 1 + \sin \left({\alpha - \beta } \right)\left[ {\sin \left({2\gamma - \alpha - \beta } \right) - 2\cos \left({\gamma - \alpha } \right)\sin \left({\gamma - \beta } \right)} \right]\\ = 1 + \sin \left({\alpha - \beta } \right)\left[ {\sin \left({2\gamma - \alpha - \beta } \right) - \sin \left({2\gamma - \alpha - \beta } \right) - \sin \left({\alpha - \beta } \right)} \right]\\ = 1 - {\sin ^2}\left({\alpha - \beta } \right) = {\cos ^2}\left({\alpha - \beta } \right)\end{array}\)
Ta có
\(\begin{array}{l}{\cos ^2}\left( {\gamma - \alpha } \right) + {\sin ^2}\left({\gamma - \beta } \right)\\ = \dfrac{{1 + \cos 2\left({\gamma - \alpha } \right)}}{2} + \dfrac{{1 - \cos 2\left({\gamma - \beta } \right)}}{2}\\ = 1 + \dfrac{1}{2}\left[ {\cos 2\left({\gamma - \alpha } \right) - \cos 2\left({\gamma - \beta } \right)} \right]\\ = 1 + \sin \left({2\gamma - \alpha - \beta } \right)\sin \left({\alpha - \beta } \right)\end{array}\)
Từ đó
\(\begin{array}{l}{\cos ^2}\left( {\gamma - \alpha } \right) + {\sin ^2}\left({\gamma - \beta } \right) - 2\cos \left({\gamma - \alpha } \right)\sin \left({\gamma - \beta } \right)\sin \left({\alpha - \beta } \right)\\ = 1 + \sin \left({2\gamma - \alpha - \beta } \right)\sin \left({\alpha - \beta } \right) - 2\cos \left({\gamma - \alpha } \right)\sin \left({\gamma - \beta } \right)\sin \left({\alpha - \beta } \right)\\ = 1 + \sin \left({\alpha - \beta } \right)\left[ {\sin \left({2\gamma - \alpha - \beta } \right) - 2\cos \left({\gamma - \alpha } \right)\sin \left({\gamma - \beta } \right)} \right]\\ = 1 + \sin \left({\alpha - \beta } \right)\left[ {\sin \left({2\gamma - \alpha - \beta } \right) - \sin \left({2\gamma - \alpha - \beta } \right) - \sin \left({\alpha - \beta } \right)} \right]\\ = 1 - {\sin ^2}\left({\alpha - \beta } \right) = {\cos ^2}\left({\alpha - \beta } \right)\end{array}\)